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A Lagrangian Based Approach for the Asymmetric Generalized Traveling Salesman Problem

Author

Listed:
  • Charles E. Noon

    (University of Tennessee, Knoxville, Tennessee)

  • James C. Bean

    (University of Michigan, Ann Arbor, Michigan)

Abstract

This paper presents an optimal approach for the asymmetric Generalized Traveling Salesman Problem (GTSP). The GTSP is defined on a directed graph in which the nodes are grouped into m predefined, mutually exclusive and exhaustive sets with the arc set containing no intraset arcs. The problem is to find a minimum cost m -arc directed cycle which includes exactly one node from each set. Our approach employs a Lagrangian relaxation to compute a lower bound on the total cost of an optimal solution. The lower bound and a heuristically determined upper bound are used to identify and remove arcs and nodes which are guaranteed not to be in an optimal solution. Finally, we use an efficient branch-and-bound procedure which exploits the multiple choice structure of the node sets. We present computational results for the optimal approach tested on a series of randomly generated problems. The results show success on a range of problems with up to 104 nodes.

Suggested Citation

  • Charles E. Noon & James C. Bean, 1991. "A Lagrangian Based Approach for the Asymmetric Generalized Traveling Salesman Problem," Operations Research, INFORMS, vol. 39(4), pages 623-632, August.
    • Handle: RePEc:inm:oropre:v:39:y:1991:i:4:p:623-632
    DOI: 10.1287/opre.39.4.623
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    Citations

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    Cited by:

    1. Jeanette Schmidt & Stefan Irnich, 2020. "New Neighborhoods and an Iterated Local Search Algorithm for the Generalized Traveling Salesman Problem," Working Papers 2020, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz.
    2. Gharehgozli, Amir & Zaerpour, Nima, 2020. "Robot scheduling for pod retrieval in a robotic mobile fulfillment system," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 142(C).
    3. Karapetyan, D. & Gutin, G., 2012. "Efficient local search algorithms for known and new neighborhoods for the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 219(2), pages 234-251.
    4. Alice E. Smith, 2023. "Note from the Editor," INFORMS Journal on Computing, INFORMS, vol. 35(4), pages 711-712, July.
    5. Bagchi, Tapan P. & Gupta, Jatinder N.D. & Sriskandarajah, Chelliah, 2006. "A review of TSP based approaches for flowshop scheduling," European Journal of Operational Research, Elsevier, vol. 169(3), pages 816-854, March.
    6. W Zahrouni & H Kamoun, 2011. "Transforming part-sequencing problems in a robotic cell into a GTSP," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(1), pages 114-123, January.
    7. John Gunnar Carlsson & Mehdi Behroozi & Raghuveer Devulapalli & Xiangfei Meng, 2016. "Household-Level Economies of Scale in Transportation," Operations Research, INFORMS, vol. 64(6), pages 1372-1387, December.
    8. Rajabighamchi, Farzaneh & van Hoesel, Stan & Defryn, Christof, 2023. "The order picking problem under a scattered storage policy," Research Memorandum 006, Maastricht University, Graduate School of Business and Economics (GSBE).
    9. Snyder, Lawrence V. & Daskin, Mark S., 2006. "A random-key genetic algorithm for the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 174(1), pages 38-53, October.
    10. Karapetyan, D. & Gutin, G., 2011. "Lin-Kernighan heuristic adaptations for the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 208(3), pages 221-232, February.
    11. Pop, Petrică C. & Cosma, Ovidiu & Sabo, Cosmin & Sitar, Corina Pop, 2024. "A comprehensive survey on the generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 314(3), pages 819-835.
    12. Gharehgozli, Amir & Yu, Yugang & de Koster, René & Du, Shaofu, 2019. "Sequencing storage and retrieval requests in a container block with multiple open locations," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 125(C), pages 261-284.
    13. Renaud, Jacques & Boctor, Fayez F., 1998. "An efficient composite heuristic for the symmetric generalized traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 108(3), pages 571-584, August.
    14. Ahmadi, Reza H. & Mamer, John W., 1999. "Routing heuristics for automated pick and place machines," European Journal of Operational Research, Elsevier, vol. 117(3), pages 533-552, September.
    15. Amir Hossein Gharehgozli & Gilbert Laporte & Yugang Yu & René de Koster, 2015. "Scheduling Twin Yard Cranes in a Container Block," Transportation Science, INFORMS, vol. 49(3), pages 686-705, August.
    16. J-Y Potvin & M-A Naud, 2011. "Tabu search with ejection chains for the vehicle routing problem with private fleet and common carrier," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(2), pages 326-336, February.
    17. M Blais & G Laporte, 2003. "Exact solution of the generalized routing problem through graph transformations," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 54(8), pages 906-910, August.
    18. Yuan, Yuan & Cattaruzza, Diego & Ogier, Maxime & Semet, Frédéric, 2020. "A branch-and-cut algorithm for the generalized traveling salesman problem with time windows," European Journal of Operational Research, Elsevier, vol. 286(3), pages 849-866.
    19. Feremans, Corinne & Labbe, Martine & Laporte, Gilbert, 2003. "Generalized network design problems," European Journal of Operational Research, Elsevier, vol. 148(1), pages 1-13, July.

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